Bayesian Parameter Estimation Method Research Articles

Efficiency Factors for Reinforced Concrete Deep Beams: Part 2 - Code Calibration

This paper presents the limit state shear design formulas for both normal and high-strength reinforced concrete deep beams using strut-and-tie model (STM). The proposed equations are based on the STMs with six state-of-the-art efficiency factors. These STMs were improved by correcting the bias and quantifying the scatter using a Bayesian parameter estimation method. The statistical parameters of material properties, dimensions, and the accuracy of design equations are considered to develop the resistance models obtained by Monte Carlo simulations. The reliability analysis is performed to determine the strength reduction factors. The calculated values of strength reduction factors are proposed for each of the considered efficiency factors.

Efficiency Factors for Reinforced Concrete Deep Beams: Part 1 - Improved Models

A strut-and-tie model (STM) has been used as a rational and simple method for analysis and design of reinforced concrete deep beams. The STM idealizes the discontinuity regions in such members as a truss-like structure consisting of compression struts and tension ties. The strength of a deep beam is usually controlled by the capacity of the diagonal concrete struts which is generally proposed in terms of the concrete strength and the efficiency factor. The efficiency factor accounts for the variation of the concrete material and the compression softening effect. One has been used in many STMs and often exhibit biases and uncertain errors in prediction of the shear strengths of deep beams. This uncertainty is due to the imperfection of formulation, missing parameters, and insufficient experimental data. Based on a database of 406 test results and six state-of-the-art formulations of the efficiency factors found in the international building codes and literature, this paper proposes improved STMs for accurate prediction of shear strength of simply supported reinforced concrete deep beams by correcting the bias and quantifying the scatter using a Bayesian parameter estimation method.

HDDM: Hierarchical Bayesian estimation of the Drift-Diffusion Model in Python

The diffusion model is a commonly used tool to infer latent psychological processes underlying decision-making, and to link them to neural mechanisms based on response times. Although efficient open source software has been made available to quantitatively fit the model to data, current estimation methods require an abundance of response time measurements to recover meaningful parameters, and only provide point estimates of each parameter. In contrast, hierarchical Bayesian parameter estimation methods are useful for enhancing statistical power, allowing for simultaneous estimation of individual subject parameters and the group distribution that they are drawn from, while also providing measures of uncertainty in these parameters in the posterior distribution. Here, we present a novel Python-based toolbox called HDDM (hierarchical drift diffusion model), which allows fast and flexible estimation of the the drift-diffusion model and the related linear ballistic accumulator model. HDDM requires fewer data per subject/condition than non-hierarchical methods, allows for full Bayesian data analysis, and can handle outliers in the data. Finally, HDDM supports the estimation of how trial-by-trial measurements (e.g., fMRI) influence decision-making parameters. This paper will first describe the theoretical background of the drift diffusion model and Bayesian inference. We then illustrate usage of the toolbox on a real-world data set from our lab. Finally, parameter recovery studies show that HDDM beats alternative fitting methods like the χ2-quantile method as well as maximum likelihood estimation. The software and documentation can be downloaded at: http://ski.clps.brown.edu/hddm_docs/

A Simplified Approach to Joint Shear Behavior Prediction of RC Beam-Column Connections

An extensive experimental database of reinforced concrete (RC) beam-column connections subjected to cyclic lateral loading has been constructed. All cases within the database experienced joint shear failure, either in conjunction with or without yielding of longitudinal beam reinforcement, representing damage within a joint panel that was the main contributor to total lateral deformation. (Cases having damage within a joint panel caused by other premature failure modes (e.g., anchorage failure) are not included in the database.) Using the experimental database, envelope curves of joint shear stress vs. strain behavior were developed by connecting key points such as cracking, yielding, and peak loading. Joint shear stress and strain models at peak response have been developed by a Bayesian parameter estimation method based on the experimental database. At other key points, important influence parameters are also identified by constructing joint shear stress and strain models in conjunction with the Bayesian parameter estimation method. Then, a complete RC joint shear stress vs. strain model (including post-peak behavior) is suggested using simplified joint shear stress and strain models at peak response; effects of key parameters on the suggested behavior models are evaluated. Finally, the ASCE/SEI 41 joint shear behavior model has been examined using the constructed database—specific joint shear strength factors and plastic joint shear deformation values are recommended for use when following that approach.

Impact on the tensor-to-scalar ratio of incorrect Galactic foreground modelling

A key goal of many Cosmic Microwave Background experiments is the detection of gravitational waves, through their B-mode polarization signal at large scales. To extract such a signal requires modelling contamination from the Galaxy. Using the Planck experiment as an example, we investigate the impact of incorrectly modelling foregrounds on estimates of the polarized CMB, quantified by the bias in tensor-to-scalar ratio r, and optical depth tau. We use a Bayesian parameter estimation method to estimate the CMB, synchrotron, and thermal dust components from simulated observations spanning 30-353 GHz, starting from a model that fits the simulated data, returning r<0.03 at 95% confidence for an r=0 model, and r=0.09+-0.03 for an r=0.1 model. We then introduce a set of mismatches between the simulated data and assumed model. Including a curvature of the synchrotron spectral index with frequency, but assuming a power-law model, can bias r high by ~1-sigma (delta r ~ 0.03). A similar bias is seen for thermal dust with a modified black-body frequency dependence, incorrectly modelled as a power-law. If too much freedom is allowed in the model, for example fitting for spectral indices in 3 degree pixels over the sky with physically reasonable priors, we find r can be biased up to ~3-sigma high by effectively setting the indices to the wrong values. Increasing the signal-to-noise ratio by reducing parameters, or adding additional foreground data, reduces the bias. We also find that neglecting a 1% polarized free-free or spinning dust component has a negligible effect on r. These tests highlight the importance of modelling the foregrounds in a way that allows for sufficient complexity, while minimizing the number of free parameters.

Bayesian multiple comparison of models for binary data with inequality constraints

In this paper we consider generalized linear models for binary data subject to inequality constraints on the regression coefficients, and propose a simple and efficient Bayesian method for parameter estimation and model selection by using Markov chain Monte Carlo (MCMC). In implementing MCMC, we introduce appropriate latent variables and use a simple approximation of a link function, to resolve computational difficulties and obtain convenient forms for full conditional posterior densities of elements of parameters. Bayes factors are computed via the Savage-Dickey density ratios and the method of Oh (Comput. Stat. Data Anal. 29:411---427, 1999), for which posterior samples from the full model with no degenerate parameter and the full conditional posterior densities of elements are needed. Since it uses one set of posterior samples from the full model for any model in consideration, it performs simultaneous comparison of all possible models and is very efficient compared with other model selection methods which require one to fit all candidate models. A simulation study shows that significant improvements can be made by taking the constraints into account. Real data on purchase intention of a product subject to order constraints is analyzed by using the proposed method. The analysis results show that there exist some price changes which significantly affect the consumer behavior. The results also show the importance of simultaneous comparison of models rather than separate pairwise comparisons of models since the latter may yield misleading results from ignoring possible correlations between parameters.

Bayesian Parameter Estimation Using Periodic Cost Functions

In this paper, a new method for Bayesian periodic parameter estimation is derived using periodic cost functions. The method, named parameter estimation via root finding (PERF), is based on Fourier series representation of the Bayes periodic-risk functions. The PERF method is implemented for minimum cyclic error, minimum absolute periodic error, and minimum mean-square-periodic-error (MSPE) estimators and the corresponding estimators are derived. The periodic estimators are applied to direction-of-arrival and phase estimation problems and compared with the minimum mean-square-error and maximum a posteriori probability estimators, and the periodic Ziv-Zakai lower bound in terms of MSPE.

Bayesian statistical analysis applied to solar radiation modelling

Abstract This paper proposes to use a rather new statistical approach in the realm of solar radiation modelling namely Bayesian inference. In this work, the theory of Bayesian inference will be presented at length. The Bayesian analysis consists in two levels. The first one is related to the parameter estimation while the second one concerns the model selection problem. As an illustration, a Bayesian parameter estimation method is used to derive a logistic hourly solar diffuse fraction model. A major difference between Bayesian and frequentist (or classical) methods is that the Bayesian inference offers a framework (through the use of prior information) to continuously update our posterior beliefs. In other words, all previous work is not wasted as the preceding model’s parameters can be used as prior information for the derivation of the parameters estimates of the next (new) model. For this particular application, it is also shown that the use of Bayesian methods instead of classical statistical techniques lead to a less biased model.

Joint Shear Behavior Prediction for RC Beam-Column Connections

An extensive database has been constructed of reinforced concrete (RC) beam-column connection tests subjected to cyclic lateral loading. All cases within the database experienced joint shear failure, either in conjunction with or without yielding of longitudinal beam reinforcement. Using the experimental database, envelope curves of joint shear stress vs. joint shear strain behavior have been created by connecting key points such as cracking, yielding, and peak loading. Various prediction approaches for RC joint shear behavior are discussed using the constructed experimental database. RC joint shear strength and deformation models are first presented using the database in conjunction with a Bayesian parameter estimation method, and then a complete model applicable to the full range of RC joint shear behavior is suggested. An RC joint shear prediction model following a U.S. standard is next summarized and evaluated. Finally, a particular joint shear prediction model using basic joint shear resistance mechanisms is described and for the first time critically assessed.

Bayesian Parameter Estimation for Characterizing the Cyclic Variation of Echocardiographic Backscatter to Assess the Hearts of Asymptomatic Type 2 Diabetes Mellitus Subjects

Previous studies have shown that effective quantification of the cyclic variation of myocardial ultrasonic backscatter over the heart cycle might provide a non-invasive technique for identifying the early onset of cardiac abnormalities. These studies have demonstrated the potential for measurements of the magnitude and time delay of cyclic variation for identifying early onset of disease. The goal of this study was to extend this approach by extracting additional parameters characterizing the cyclic variation in an effort to better assess subtle changes in myocardial properties in asymptomatic subjects with type 2 diabetes. Echocardiographic images were obtained on a total of 43 age-matched normal control subjects and 100 type 2 diabetics. Cyclic variation data were generated by measuring the average level of ultrasonic backscatter over the heart cycle within a region of interest placed in the posterior wall of the left ventricle. Cyclic variation waveforms were modeled as piecewise linear functions, and quantified using a novel Bayesian parameter estimation method. Magnitude, rise time and slew rate parameters were extracted from models of the data. The ability of each of these parameters to distinguish between normal and type 2 diabetic subjects, and between subjects grouped by glycated hemoglobin (HbA1c) was compared. Results suggest a significant improvement in using measurements of the rise time and slew rate parameters of cyclic variation to differentiate ( P < 0.001) the hearts of patients segregated based on widely employed indices of diabetic control compared to differentiation based on the magnitude of cyclic variation.(E-mail: james.g.miller@wustl.edu)

CONSTRAINTS ON COSMIC-RAY PROPAGATION MODELS FROM A GLOBAL BAYESIAN ANALYSIS

Research in many areas of modern physics such as, e.g., indirect searches for dark matter and particle acceleration in SNR shocks, rely heavily on studies of cosmic rays (CRs) and associated diffuse emissions (radio, microwave, X-rays, gamma rays). While very detailed numerical models of CR propagation exist, a quantitative statistical analysis of such models has been so far hampered by the large computational effort that those models require. Although statistical analyses have been carried out before using semi-analytical models (where the computation is much faster), the evaluation of the results obtained from such models is difficult, as they necessarily suffer from many simplifying assumptions, The main objective of this paper is to present a working method for a full Bayesian parameter estimation for a numerical CR propagation model. For this study, we use the GALPROP code, the most advanced of its kind, that uses astrophysical information, nuclear and particle data as input to self-consistently predict CRs, gamma rays, synchrotron and other observables. We demonstrate that a full Bayesian analysis is possible using nested sampling and Markov Chain Monte Carlo methods (implemented in the SuperBayeS code) despite the heavy computational demands of a numerical propagation code. The best-fit values of parameters found in this analysis are in agreement with previous, significantly simpler, studies also based on GALPROP.

CHARACTERIZING THE CONTAMINATING DISTANCE DISTRIBUTION FOR BAYESIAN SUPERNOVA COSMOLOGY

Measurements of the equation of state of dark energy from surveys of thousands of Type Ia Supernovae will be limited by spectroscopic follow-up and must therefore rely on photometric identification, increasing the chance that the sample is contaminated by core collapse supernovae (CC SNe). Bayesian methods for SN cosmology can remove contamination bias while maintaining high statistical precision but are sensitive to the choice of parameterization of the contaminating distance distribution. We use simulations to investigate the form of the contaminating distribution and its dependence on the absolute magnitudes, light curve shapes, colors, extinction, and redshifts of CC SNe. We find that the CC luminosity function (LF) dominates the distance distribution function, but its shape is increasingly distorted as the redshift increases and more CC SNe fall below the survey magnitude limit. The shapes and colors of the CC light curves generally shift the distance distribution, and their effect on the CC distances is correlated. We compare the simulated distances to the first year results of the SDSS-II SN survey and find that the SDSS distance distributions can be reproduced with simulated CC SNe that are ∼1 mag fainter than the standard Richardson et al. LFs, which do not produce a good fit. To exploit the full power of the Bayesian parameter estimation method, parameterization of the contaminating distribution should be guided by the current knowledge of the CC LFs, coupled with the effects of the survey selection and magnitude limit, and allow for systematic shifts caused by the parameters of the distance fit.

Transition of water demand and treatment plant overcapacity: a case study of the Seoul Metropolitan City, Korea

This study deals with the overcapacity problem of water treatment plants in Korea, and mainly discusses status, causes, and engineering options. To this end, we first statistically analyze the recent trend of demand, revealing that the demands of small- and mid-size systems are still increasing while that of large-size systems is now decreasing. Since the existing approach to plan capacity implicitly assumes that demand will increase at a regular rate, we estimate excess capacities and system utilizations of large-size systems. From these results it is found that the large-size systems are suffering from serious overcapacity, thus necessitating that engineers make very difficult decisions given that systems are still expanding the capacities of plants due to a lack of awareness of the current demand trend. For other systems where there is a better understanding of the transition of demand, planners have ceased to expand plants or have closed down relatively old plants in efforts to reduce O&M costs. To address this problem, quick recognition of the transition of demand is being highlighted by the concepts of integrated resources management and cybernetics. Therefore, we examined how quickly the new trend of the Seoul case could be precisely recognized and appropriately addressed. Using the Bayesian parameter estimation method, we found that a new trend can be recognized six years after the transition of demand.

Probabilistic shear strength models for reinforced concrete beams without shear reinforcement

In order to predict the shear strengths of reinforced concrete beams, many deterministic models have been developed based on rules of mechanics and on experimental test results. While the constant and variable angle truss models are known to provide reliable bases and to give reasonable predictions for the shear strengths of members with shear reinforcement, in the case of members without shear reinforcement, even advanced models with complicated procedures may show lack of accuracy or lead to fairly different predictions from other similar models. For this reason, many research efforts have been made for more accurate predictions, which resulted in important recent publications. This paper develops probabilistic shear strength models for reinforced concrete beams without shear reinforcement based on deterministic shear strength models, understanding of shear transfer mechanisms and influential parameters, and experimental test results reported in the literature. Using a Bayesian parameter estimation method, the biases of base deterministic models are identified as algebraic functions of input parameters and the errors of the developed models remaining after the bias-correction are quantified in a stochastic manner. The proposed probabilistic models predict the shear strengths with improved accuracy and help incorporate the model uncertainties into vulnerability estimations and risk-quantified designs.

Joint shear behaviour of reinforced concrete beam–column connections

An extensive experimental database (341 cases in total) has been constructed for reinforced concrete (RC) beam–column connections subjected to quasi-static cyclic lateral loading. All cases within the database experienced joint shear failure, either in conjunction with or without yielding of beam reinforcement. RC joint shear strength models, which are applicable to both modern and older types of beam–column connections, have been developed using the constructed database in conjunction with a Bayesian parameter estimation method. Following this same procedure established to develop the strength models, joint shear deformation models (at maximum joint shear stress) have also been developed. The proposed quantitative RC joint shear strength and deformation models indicate that the following parameters are most important towards determining joint shear strength and deformation capacity–concrete compressive strength, beam reinforcement, joint transverse reinforcement, in-plane geometry, out-of-plane geometry and joint eccentricity. Finally, the influence of these six key parameters on the RC joint shear stress against joint shear strain behaviour at peak response has been assessed.

Bayesian and maximum entropy methods for fusion diagnostic measurements with compact neutron spectrometers

In applications of neutron spectrometry to fusion diagnostics, it is advantageous to use methods of data analysis which can extract information from the spectrum that is directly related to the parameters of interest that describe the plasma. We present here methods of data analysis which were developed with this goal in mind, and which were applied to spectrometric measurements made with an organic liquid scintillation detector (type NE213). In our approach, we combine Bayesian parameter estimation methods and unfolding methods based on the maximum entropy principle. This two-step method allows us to optimize the analysis of the data depending on the type of information that we want to extract from the measurements. To illustrate these methods, we analyze neutron measurements made at the PTB accelerator under controlled conditions, using accelerator-produced neutron beams. Although the methods have been chosen with a specific application in mind, they are general enough to be useful for many other types of measurements.

Latent Function Sigmoid Compression Bayesian Fault Recognition Method

Support vector machines(SVM)are extensively exploited for small sample fault recognition.On the contrary,due to difficulties such as a priori knowledge acquisition and nonlinear recognition,fault recognition approaches based on Bayesian probability have found few applications.To deal with these problems,the latent function sigmoid compression Bayesian fault recognition(LS-BFR)is proposed.Based on Gaussian stochastic process and Bayesian probability,LS-BFR takes Gaussian regression as the latent function,and changes the regression output into probability by sigmoid compression.To improve the performance of LS-BFR for nonlinear fault recognition,kernel functions are introduced to perform latent Gaussian process regression, and a practical approach based on the Bayesian parameter estimation method is proposed to determine kernel function parameters. The performance of LS-BFR is validated by recognizing the faults of misalignment and unbalance simulated on a rotor testbed. Experiment results show that the LS-BFR approach based on latent function and S-compression can effectively perform small sample fault recognition and achieve better fault recognition accuracy than SVM.

Probabilistic Joint Shear Strength Models for Design of RC Beam-Column Connections

This paper constructs an extensive experimental database that includes all the types of reinforced concrete (RC) beam-column connections found in the literature to have experimental joint shear failure. Three kinds of RC joint shear strength models are developed using this database in conjunction with a Bayesian parameter estimation method. The improved performance of the developed joint shear strength models is evaluated by comparison with other deterministic RC joint shear strength models from the literature and codes. A modified ACI 352R-02 joint shear strength definition for design, with adjusted joint shear strength factors for greater consistency, is suggested.

A new statistical approach for joint shear strength determination of RC beam-column connections subjected to lateral earthquake loading

Reinforced concrete (RC) joint shear strength models are constructed using an experimental database in conjunction with a Bayesian parameter estimation method. The experimental database consists of RC beam-column connection test subassemblies that maintained proper confinement within the joint panel. All included test subassemblies were subjected to quasi-static cyclic lateral loading and eventually experienced joint shear failure (either in conjunction with or without yielding of beam reinforcement); subassemblies with out-of-plane members and/or eccentricity between the beam(s) and the column are not included in this study. Three types of joint shear strength models are developed. The first model considers all possible influence parameters on joint shear strength. The second model contains those parameters left after a step-wise process that systematically identifies and removes the least important parameters affecting RC joint shear strength. The third model simplifies the second model for convenient application in practical design. All three models are unbiased and show similar levels of scatter. Finally, the improved performance of the simplified model for design is identified by comparison with the current ACI 352R-02 RC joint shear strength model.

A Modified Extended Bayesian Method for Parameter Estimation

This paper presents a modified extended Bayesian method for parameter estimation. In this method the mean value of the a priori estimation is taken from the values of the estimated parameters in the previous iteration step. In this way, the parameter covariance matrix can be automatically updated during the estimation procedure, thereby avoiding the selection of an empirical parameter. Because the extended Bayesian method can be regarded as a Tikhonov regularization, this new method is more stable than both the least-squares method and the maximum likelihood method. The validity of the proposed method is illustrated by two examples: one based on simulated data and one based on real engineering data.